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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E2:10"/><i><font color="Green">E9</font></i>: 
 <a href="binarith.html#K11">add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binari_4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E11</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K11">add_ovfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="binari_4.html#E2:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<i><font color="Green">E12</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K10">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K11">add_ovfl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T47">BINARITH:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E3:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="binari_4.html#E3:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
;<br/>
<a NAME="E6:10"/><i><font color="Green">E13</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E7:10"/>
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K10">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_4.html#E1:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="binari_4.html#E4:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="binari_4.html#E5:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="binari_4.html#E6:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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